The cores of most viruses have inherent negative charges due to the phosphate group in genomes (Westheimer, F. H. Why nature chose phosphates. Science 235, 1173-1178 (1987)). In contrast, the amino acids of viral capsids or envelopes have complex charge distributions on the surface of viruses. Such core-shell charge separation paves the road for the dipolar coupling between electromagnetic waves and confined acoustic vibrations. The MRA processes should thus occur in viruses when the confined acoustic vibration moves the charges and changes their dipole moments. For spherical viruses, such a vibrational mode is the spheroidal (SPH) mode with an angular momentum l=1. (Lamb, H. On the vibrations of an elastic sphere. Proc. London Math. Soc. 13, 189-212 (1882), Duval, E. Far-infrared and Raman vibrational transitions of a solid sphere: Selection rules. Phys. Rev. B 46, 5795-5797 (1992)) Due to its Raman inactive nature, the l=1 SPH mode has been rarely studied experimentally, for example, there is no published report on its measured oscillation frequency versus size. On the other hand, both the theoretically calculated [SPH, l=1, n=0] and [SPH, l=1, n=1] dipolar modes (See FIG. 1a) have relative displacement between the core (indicated by light-colored arrows) and shell (indicated by dark-colored arrows) (Murray, D. B. et al. Far-infrared absorption by acoustic phonons in Titanium dioxide nanopowders. J. Nanoelectron. Optoelectron. 1, 92-98 (2006)), which should induce MRA through the core-shell charge structure of a nanoparticle, including viruses. By assuming a virus as a homogeneous sphere with elastic parameters close to the Satellite Tobacco Mosaic Virus crystal, (Stephanidis, B., Adichtchev, S., Gouet, P., McPherson, A. Mermet, A. Elastic properties of viruses. Biophys. J. 93, 1354-1359 (2007)) one can estimate the frequency of the [SPH, l=1, n=0] mode for a 30 nm virus to be around 40 GHz. The frequencies of the dipolar modes are determined by the longitudinal sound velocity VL, transverse sound velocity VT, and the radius of viruses R. (Lamb, H. On the vibrations of an elastic sphere. Proc. London Math. Soc. 13, 189-212 (1882) Assuming VL/VT18 2 for viruses (Talati, M. & Jha, K. Acoustic phonon quantization and low-frequency Raman spectra of spherical viruses. Phys. Rev. E 73, 011901 (2006)), the eigen frequencies are expected to be proportional to VL/R For most viruses, their sizes range from 10 nm to 300 nm and their VL doesn't change too much, thus making the frequency of the dipolar modes falls in the microwave range.
Except for spheres, rods and filaments are among the other shapes most commonly found in viruses. The charge separation between the core and two tips could also induce MRA related to the longitudinal acoustic standing waves. However, different from the SPH confined acoustic modes in a sphere, not all orders of the confined longitudinal acoustic waves in a rod-like structure can induce MRA. FIG. 1b illustrates that only standing waves with the mode order N equal to 2(2m+1) have relative motion between two ends and the center of the rod, where m=0, 1, 2 . . . . It is expected that the selection rule of the high-order-acoustic-mode absorption varies with the shape of viruses.
Besides, the homogeneous broadening linewidth of the MRA can further provide information regarding the mechanical properties of the virus. Considering small acoustic impedance contrast (Westheimer, F. H. Why nature chose phosphates. Science 235, 1173-1178 (1987)) between water and a virus, the theoretically predicted mechanical quality factor of the viral confined acoustic modes is much less than one (Talati, M., Jha, P. K. Acoustic phonon quantization and low-frequency Raman spectra of spherical viruses. Phys. Rev. E 73, 011901 (2006)). In contrast, the standard deviation of the viral size is typically <±5%, which makes the MRA quality factor QMRA larger than 10. Therefore, the microwave spectral linewidth of a virus should be dominated by the homogeneous broadening, making MRA bandwidth another useful indicator for the mechanical properties of viruses.